Problems

Detective Nero Wolf investigates a crime. He’s got $80$ people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these $80$ people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in $12$ days?

The king decided to reward a group of $n$ wise men. They will be placed in a row one after the other (so that everyone is looking in the same direction), and each is going to wear a black or a white hat. Everyone will see the hats of everyone in front, but not those behind them. The wise men will take turns (from the last to the first) to name the color (white or black) and the natural number of their choice.
At the end, the number of sages who have named the color of their hat correctly is counted: that is exactly how many days the whole group will be paid a salary raise. The wise men were allowed to agree in advance on how to respond. At the same time, the wise men know that exactly $k$ of them are insane (they do not know who exactly). Any insane man names the color white or black, regardless of the agreement. What is the maximum number of days with a pay supplement that the wise men can guarantee to a group, regardless of the location of the insane in the queue?

A whole number of litres of water were poured into three vessels. You can only to pour into any vessel the exact amount of water equal to the amount it already contains from any other vessel. Prove that in a few transfusions one can empty one of the vessels. The vessels are large enough: each can hold all the water.

A set includes weights weighing $1$ gram, $2$ grams, $4$ grams, ... (all powers of the number $2$), and in the set some of the weights might be the same. Weights were placed on two cups of the scales so that the scales are in balance. It is known that on the left cup, all weights are different. Prove that there are as many weights on the right cup as there are on the left.

There are $16$ cities in the kingdom. Prove that it is possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most five roads coming out of each city.

Show that a bipartite graph with $n$ vertices cannot have more than $\frac{n^2}{4}$ edges.

In a graph $G$, we call a matching any choice of edges in $G$ in such a way that all vertices have only one edge among chosen connected to them. A perfect matching is a matching which is arranged on all vertices of the graph.
Let $G$ be a graph with $2n$ vertices and all the vertices have degree at least $n$ (the number of edges exiting the vertex). Prove that one can choose a perfect matching in $G$.

A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?

Now imagine you got $10$ new guests arriving to the completely full hotel. What should you do now?

The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers $1,2,3...$ corresponding to all natural numbers. How to deal with this one?